36 XI! TOPOLOGY AND TOPOLOGICAL ALGEBRA (12.8.4) For a homomorphism f of a topological group G into a topological group G' to be continuous, it is necessary and sufficient that f should be con- tinuous at one point. For if /is continuous at a e G and if V is a neighborhood off(a), then f~l(V) = V is a neighborhood of a. For each x e G, we have which establishes the continuity of /at the point x, by virtue of (12.8.3(i)). If H is a subgroup of a topological group G, the induced topology on H is clearly compatible with the group structure of H. Whenever we consider H as a topological group, it is to be understood in this sense unless the contrary is explicitly stated. (12.8.5) The closure H of a subgroup (resp. normal subgroup) H of a topologi- cal group G is a subgroup (resp. normal subgroup) ofG.IfG is Hausdorff and H is commutative, then H is commutative. The image ofHxH=HxH under the continuous mapping (x, y) i-» xy 1 of G x G into G is contained in H, because the image of H x H under this mapping is contained in H (3.11.4). Hence H is a subgroup. Likewise, if H is normal and a e G, the image of H under the mapping x\-+ axa~~1 is contained in H, hence the image of H is contained in H, and therefore H is normal. Finally, if G is Hausdorff and H is commutative, the continuous functions xy and yx are equal on H x H and therefore also on H x fl by virtue of the principle of extension of identities ((3.15.2), 12.3, and 12.5). (12.8.6) (i) In a topological group G, the normalizer Jf (H) of a closed sub- group H (i.e., the set of all x e G such that xHx"1 c H) is a closed subgroup. (ii) In a Hausdorff topological group G, the centralizer «$?(M) of any subset M ofG (i.e., the set of all x e G which commute with every element of M) is a closed subgroup ofG. In particular, the center ofG is closed. For each z e H, the set of elements x e G such that xzx"1 e H is the inverse image of H under the continuous mapping xh-+xzx"~19 hence is a closed set (3.11.4). Since ^T(H) is the intersection of these sets as z runs through H, it follows that ^T(H) is closed (3.8.2). Again, if G is Hausdorff then for each z e M the set of x e G such that zx = xz is closed (12.3.5) and hence so is Jf(M) since it is the intersection of these sets. (12.8.7) (i) In a topological group G, every locally closed subgroup is closed. Every subgroup which has an interior point is both open and closed. (ii) In a Hausdorff group, every discrete subgroup is closed.of the